Interacting telegraph particles with mutual attraction in a Markov environment in the real line

Publications on the study of the telegraph process are mainly devoted to various generalizations of this process either to multidimensional spaces or to various non-Markov environments in which this process evolves. Only a few works are devoted to the study of two or more telegraph processes that describe the evolution of interacting telegraph particles. In this paper, the characteristics of the time of the first collision of two particles with a shift toward each other are studied. We also study the number of collisions of n telegraph particles with a shift toward each other. Such a system can be an approximate model for the gravitational or electromagnetic interaction of two telegraph particles in a Markov environment.